Are the diagonals of a parallelogram perpendicular? Why or why not? Explain.
Does an isosceles trapezoid have two sets of parallel sides? Why or why not? Explain.
Does a rhombus have two sets of congruent sides? Why or why not? Explain.
Answer: 3. Yes, because the diagonals of a rhombus, which is a parallelogram, are perpendicular. In a parallelogram diagonals bisect each other.
4. No, because in an isosceles trapezoid the sides that are not parallel are equal.
5. Yes, because a rhombus has two sets of parallel sides and all sides are congruent.
To answer questions regarding properties of geometric shapes like parallelograms, trapezoids, and rhombuses, we need to understand their definitions and properties.
1. Are the diagonals of a parallelogram perpendicular? Why or why not? Explain.
A parallelogram is a quadrilateral with both pairs of opposite sides parallel. The diagonals of a parallelogram are the line segments connecting the opposite corners.
To determine if the diagonals are perpendicular, we can consider two cases:
a) If the angles of the parallelogram are right angles (90 degrees), then the diagonals will be perpendicular. This case corresponds to a special type of parallelogram called a rectangle, where all angles are right angles.
b) If the angles of the parallelogram are not right angles, then the diagonals will not be perpendicular. In this case, the diagonals will intersect at a point within the parallelogram, dividing each other into equal segments. However, they will not form right angles with each other.
So, the diagonals of a parallelogram can be perpendicular in certain cases (when the angles are right angles), but they are not always perpendicular.
2. Does an isosceles trapezoid have two sets of parallel sides? Why or why not? Explain.
An isosceles trapezoid is a trapezoid in which the non-parallel sides are congruent. A trapezoid is a quadrilateral with exactly one pair of parallel sides.
In an isosceles trapezoid, one pair of sides are parallel by definition. However, the other pair of sides (the non-parallel sides) are not parallel. They are equal in length, but their slopes are different, causing them to intersect at an angle.
Therefore, an isosceles trapezoid does not have two sets of parallel sides, but only one pair of parallel sides.
3. Does a rhombus have two sets of congruent sides? Why or why not? Explain.
A rhombus is a quadrilateral with all four sides of equal length. Additionally, a rhombus has opposite sides parallel to each other.
Therefore, a rhombus does have two sets of congruent sides. The opposite sides of a rhombus are both parallel and equal in length. This symmetry allows a rhombus to have two sets of congruent sides.
I hope this explanation helps clarify the properties of parallelograms, trapezoids, and rhombuses.